5 Number Summary Calculator

What the five number summary calculator does

The five number summary describes a data set with just five values: the minimum, the first quartile, the median, the third quartile, and the maximum. Together they capture where the data starts, where it ends, where its middle sits, and how its central half is spread. This calculator finds all five from your numbers and shows the sorted data behind them.

It is the most compact honest picture of a data set, and it is exactly what a box plot is drawn from. Below is how each of the five is found.

How to use it

1. Enter your dataset in the box, separated by commas, spaces, or new lines.
2. Press Calculate for the five number summary and your sorted data, or Reset to clear it.

How the five numbers are found

Three of the five are quick. The minimum and maximum are simply the smallest and largest values once the data is sorted, and the median is the value in the middle. The other two, the first and third quartiles, come from splitting the data at the median and finding the middle of each half.

The first quartile is the median of the lower half, and the third quartile is the median of the upper half. When the data has an odd number of values, the middle value is set aside before each half is taken. From these five, one more useful number falls out: the interquartile range, the third quartile minus the first, which measures the spread of the central half.

The five number summary and the box plot

A box plot is the five number summary drawn to scale. The box stretches from the first quartile to the third, with a line inside it at the median, so the box itself is the middle half of the data. From each end of the box a line, called a whisker, reaches out toward the minimum and the maximum.

That is why these five numbers are worth having together. In one glance they show the centre, the spread of the middle, and the full reach of the data, and they do it without being thrown off by a single extreme value the way the mean and standard deviation can be.

A worked example

Take the nine numbers 3, 5, 7, 8, 12, 13, 14, 18, 21, already sorted. The minimum is 3 and the maximum is 21. The median is the middle value, 12.

Setting that middle value aside, the lower half is 3, 5, 7, 8, whose median is (5 + 7) ÷ 2 = 6, the first quartile. The upper half is 13, 14, 18, 21, whose median is (14 + 18) ÷ 2 = 16, the third quartile. So the five number summary is 3, 6, 12, 16, and 21, and the interquartile range is 16 minus 6 = 10.

Entering your data, and the rounding

You can separate your numbers with commas, spaces, or new lines, in any mix, and the calculator sorts them for you, so the order does not matter. You need at least two values for a summary to mean anything. As with any quartile calculation, the first and third quartiles depend on the method used to split the halves, so a summary from another tool may differ a little at those two points.

Questions people ask

What is the five number summary?

A set of five values that describe a data set: the minimum, first quartile, median, third quartile, and maximum. They show the centre, the spread of the middle half, and the full range.

Why these five numbers?

Because together they give a full, outlier-resistant picture of the data, and they are exactly what a box plot is drawn from.

How do I get the interquartile range from it?

Subtract the first quartile from the third. That span, the interquartile range, measures how spread out the central half of the data is.

How does it relate to a box plot?

The box runs from the first to the third quartile with a line at the median, and the whiskers reach toward the minimum and maximum. The box plot is the five number summary drawn to scale.

References

A quick note on where the methods here come from. The five number summary and its link to the box plot are set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook that covers the summary and box plots.

1. NIST/SEMATECH e-Handbook of Statistical Methods (quartiles, box plots, and the five number summary). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (box plots and the five number summary). https://openstax.org/details/books/introductory-statistics-2e